The icosidodecadodecahedron is the uniform polyhedron with Maeder index 44 (Maeder 1997), Wenninger index 83 (Wenninger 1989), Coxeter
index 56 (Coxeter et al. 1954), and Har'El index 49 (Har'El 1993). It has
Wythoff symbol and its faces are .
Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy.
Soc. London Ser. A246, 401-450, 1954.Har'El, Z. "Uniform
Solution for Uniform Polyhedra." Geometriae Dedicata47, 57-110,
1993.Maeder, R. E. "44: Icosidodecadodecahedron." 1997.
https://www.mathconsult.ch/static/unipoly/44.html.Wenninger,
M. J. "Icosidodecadodecahedron." Model 83 in Polyhedron
Models. Cambridge, England: Cambridge University Press, pp. 128-129,
1989.
and its faces are 
.
"Coxeter",
56
],
UniformPolyhedron[
"Kaleido",
49
],
UniformPolyhedron[
"Uniform",
44
],
or UniformPolyhedron[
"Wenninger",
83
].
It is implemented in the Wolfram Language
as PolyhedronData["Icosidodecadodecahedron"].
